Journal of Chemica Acta 2 (2013) 70-72
Journal of Chemica Acta j o u r n a l h o m e p a g e : w w w . j c h e ma c t a . c o m
Zagreb Indices and Zagreb Polynomials of Polycyclic Aromatic Hydrocarbons PAHs Mohammad Reza Farahani a* a
Department of Applied Mathematics, Iran University of Science and Technology (IUST), Narmak, Tehran 16844, Iran
ARTICLE INFO
ABSTRACT
Article history: Received 3 February 2013 Received in revised form 23 February 2013 Accepted 26 February 2013 Available online 28 February 2013
Topological indices are numerical parameters of a molecular graph G which characterize its topology. There exits many structure in graph theory, with applied in chemical and nano science and vice versa and computing the connectivity indices of molecular graphs is an important branch in chemical graph theory. In this paper, we compute First Zagreb index Zg1(G)=
Keywords: Molecular graph Hydrocarbon Polycyclic Aromatic Hydrocarbons Zagreb index Zagreb polynomial
∑
uv ∈E (G )
polynomials
Zg1(G,x)=
uv ∈E (G )
x d u + dv
and
Zg2(G,x)=
∑
∑
uv ∈E (G )
uv ∈E (G )
(d u × d v )
x d u dv
and their
of a family of
© 2013 Journal of Chemica Acta
Let G be a simple connected molecular graph in chemical graph theory, then its vertices correspond to the atoms and the edges to the bonds. Throughout this paper, we denoted The vertex set, edge set, the number of vertices, edges and the degree of a vertex v (the number of its joining vertices) of a molecular graph G by V(G), E(G), υ, e and dv, respectively [1, 2]. In chemical graph theory, the topological indices of a molecular graph G are a number relation to the structure of G and are invariant on the automorphism of the graph. Chemical graph theory is an important branch of graph theory, such that there exits many topological indices in it and computing topological indices of molecular graphs is an important branch of chemical graph theory. An important topological index introduced more than forty years ago by I. Gutman and N. Trinajstic is the First Zagreb index Zg1(G) [3-4]. It is defined as the sum of squares of the vertex degrees du and dv of vertices u and v in G. Recently, we know Second Zagreb index Zg2(G) [5]. First Zagreb index Zg1(G) and Second Zagreb index Zg2(G) formulated as follow:
= Zg 2 (G )
∑
Second Zagreb index Zg2(G)=
hydrocarbon structures "Polycyclic Aromatic Hydrocarbons (PAHs)".
1. Introduction
Zg 1 (G ) =
(d u + dv ),
∑ (d u + d v )
e= uv ∈E (G )
∑ (d
e= uv ∈E (G )
u
× dv )
Also, we know two polynomials of two above indices. These are First Zagreb polynomial Zg1(G,x) and Second Zagreb polynomial Zg2(G,x) and are equal as follow
Zg 1 (G , x ) =
∑
x d u + dv
∑
x d u dv
e= uv ∈E (G )
Zg 2 (G , x ) =
e= uv ∈E (G )
We can obtain First Zagreb index Zg1(G) and Second Zagreb index Zg2(G) from its polynomial, because for i=1,2
Zg i (G ) =
∂Zg i (G , x ) |x =1 ∂x
Some basic properties of Zg1(G) can be found in some recent papers [3-13]. The goal of this paper is computing the first Zagreb index, second Zagreb index, first Zagreb polynomial and second Zagreb polynomial of a family of hydrocarbon structures “Polycyclic Aromatic Hydrocarbons (PAH)". 2. Main Results and discussion Polycyclic Aromatic Hydrocarbons PAHn is a family of hydrocarbon molecules, such that its structure is consisting of cycles with length six (Benzene). Polycyclic Aromatic Hydrocarbons can be thought as small pieces of graphene sheets with the free valences of the dangling bonds saturated by H. Vice versa, a graphene sheet can be interpreted as an infinite PAH molecule. Successful
* Corresponding author. Tel.:+98- 9192478265; e-mail:
[email protected]
© 2013 Journal of Chemica Acta ISSN: 2314-7083, Jabir Ibn Hayyan Publishing Ltd. All rights reserved
Farahani / Adv. Mat. Corrosion 1 (2013) 70-72
71 utilization of PAH molecules in modeling graphite surfaces has been reported earlier [14-27] and references therein. In this paper, we denote the first members of this hydrocarbon family as follow and are shown in Figure 1: PAH1 be the Benzene with six carbon (C) and six hydrogen (H) atoms, PAH2 be the Coronene with 24 carbon and twelve hydrogen atoms, PAH3 be the Circumcoronene with 54 carbon and eighteen hydrogen atoms.
•
For every e=uv belong to E6, du =dv=3.
E6 = E9* ={e =CC ∈ E ( PAH n ) | dC + dC =6} Now, by use above definition and above notations, we have main results of this paper in following theorem. Theorem 1 Consider the polycyclic aromatic hydrocarbons
( n∈ ) , then
molecules PAHn •
The first Zagreb polynomial of PAHn is equal to Zg1(PAHn,x)=(24pq-4p)x6+(8p)x5+px4
So the first Zagreb index of PAHn is Zg1(PAHn)=144pq+20p. •
The second Zagreb polynomial of PAHn is equal to Zg2(PAHn,x)=(24pq-4p)x9+(8p)x6+px4
So the second Zagreb index of PAHn is Zg1(PAHn)=216pq+16p. Figure 1. The first three graphs of polycyclic aromatic hydrocarbon PAHn. It is easy to see that the general representation of polycyclic aromatic hydrocarbon PAHn has 6n2 carbon (C) atoms and 6n hydrogen (H) atoms. This polycyclic aromatic hydrocarbons (or PAH family) are very similar properties to one of famous family of Benzenoid system (Circumcoronene Homologous Series of Benzenoid). The properties and applications of Benzenoid system are presented in many papers; reader can see references [28-52]. Now, before counting our favorites topological indices and polynomials of polycyclic aromatic hydrocarbon PAHn, we need present the following definition and some notations. Definition 1 [13] Consider an arbitrary vertex v with degree dv of simple connected graph G=(V(G);E(G)) and we denoted the minimum degree = with δ Min {d v | v ∈V (G )} and the maximum degree= with ∆ Max {d v | v ∈V (G )}. By according to the vertices degree, we have several partitions of vertex set V(G) and edge set E(G) of graph G, as follow
∀i ,
2δ ≤ i ≤ 2∆,
E i = {e = uv ∈ E (G ) | dv + d u = i },
∀j ,
δ 2 ≤ j ≤ ∆2 ,
E *j= {e= uv ∈ E (G ) | d v × d u= j }
δ ≤ k ≤ ∆,
∀k ,
V k= {v ∈V (G ) | d v= k }.
polycyclic aromatic hydrocarbon. This molecular graph has 6n2+6n vertices/atoms (=|V(PAHn)|) such that 6n2 of them are carbon atoms and also 6n of them hydrogen atoms. Thus, the number of edge in this hydrocarbon molecule (chemical bonds) is equal to: |E(PAHn)|=
edges of E6 (or
i =δ
Also, the edge set of graph G can be dividing to two partitions, •
E
and E6 (or
E 9* ) by yellow color in Figure 2, then we have
the number of |E4|=6n and |E6|=9n2-3n members edges of edge sets E4 (or
E3* ), and E6 (or E 9* ) of polycyclic aromatic
hydrocarbon PAHn, respectively. By according to the definition of First Zagreb polynomial and Second Zagreb polynomial, thus
∑
Zg1(PAHn,x)=
x du + d v
e∈E ( PAH n )
∑x +∑x
=
6
e∈E6
4
e∈E4
2
=(9n -3n)x +(6n) 4 And Zg2(PAHn,x)=
=
From figures, one can to see that all Carbon atoms of PAHn have degree three dCarbon=3 (or simply dC) and obviously for Hydrogen atoms dHydrogen=1. Thus, we have two vertex partitions V 3 = {v ∈V (G ) | dv = 3}, and V 1 = {v ∈V (G ) | dv = 1}. e.g. E4 (or
E3* ) by blue color and the
6
∑
x du d v
e∈E ( PAH n )
(G ) = V i .
* 3)
3 × 6n 2 +1× 6n =9n2+3n 2
Now, if we mark the edges of E4 (or
∆
such that V
( n∈ ) , be the general representation of this
Proof. Let PAHn
* 9)
E
as follow
For every e=uv belong to E4, du=3 and dv=1.
E4 = E3* ={e = HC ∈ E ( PAH n ) | d H + dC =4}
∑x +∑x 9
e∈E9* 2
4
e∈E4*
9
=(9n -3n)x +(6n) 4 Now, the first Zagreb index and Second Zagreb index of polycyclic aromatic hydrocarbon PAHn will be Zg1(PAHn)=
∂Zg1 ( PAH n , x) ∂x x =1
=(9n2-3n)×6+ 6n×4 =54 n2+6n
Farahani / Adv. Mat. Corrosion 1 (2013) 70-72
72 And Zg2(PAHn)=
∂Zg 2 ( PAH n , x) ∂x x =1
=(9n2-3n)×9+ 6n×4 =81n2-3n Here, we complete the proof of Theorem 1. □ 3. Conclusion In this paper, we count two topological indices and their topological polynomials of a family of hydrocarbon molecular graphs "polycyclic aromatic hydrocarbon PAHn". These topological indices are useful for surveying the connected structure of these nanotubes, which have relation with degrees of its vertices. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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